Lattice Study of the Jet Quenching Parameter
Marco Panero,1,2,*Kari Rummukainen,2,† and Andreas Schäfer3,‡ 1Instituto de Física Téorica UAM/CSIC, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
2
Department of Physics and Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, FI-00014, Finland
3
Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany (Received 16 December 2013; published 25 April 2014)
We present a first-principle computation of the jet quenching parameter, which describes the momentum broadening of a high-energy parton moving through the deconfined state of QCD matter at high temperature. Following an idea originally proposed by Caron-Huot, we explain how one can evaluate the soft contribution to the collision kernel characterizing this real-time phenomenon, analyzing certain gauge-invariant operators in a dimensionally reduced effective theory (electrostatic QCD), which can be studied nonperturbatively via simulations on a Euclidean lattice. Our high-precision numerical computations at two different temperatures indicate that soft contributions to the jet quenching parameter are large. After discussing the systematic uncertainties involved, we present a quantitative estimate for the jet quenching parameter in the temperature range accessible at heavy-ion colliders, and compare it to results from phenomenological models as well as to strong-coupling computations based on the holographic correspondence.
DOI:10.1103/PhysRevLett.112.162001 PACS numbers: 12.38.Gc, 11.10.Wx, 12.38.Mh
Introduction.—As first proposed by Bjorken [1], jet quenching provides a very important experimental signa-ture for the creation of the quark-gluon plasma (QGP) in heavy-ion collisions [2]. When a hard parton propagates through the deconfined medium, multiple interactions with the QGP constituents induce energy loss and momentum broadening. This leads to suppression of back-to-back correlations among final-state hadrons, and of particle yields at large transverse momenta.
A first-principle, quantitative theoretical description of this phenomenon is, however, very challenging, given that both perturbative and nonperturbative dynamics is involved [3]. The momentum broadening of a parton can be described in terms of a phenomenological parameter ˆq, defined as the average increase in the squared transverse momentum component per unit length [4]. This quantity can be computed as the second moment of the differential transverse collision rate Cðp⊥Þ describing the interaction between the hard parton (in the eikonal approximation) and the plasma constituents:
ˆq ¼hp2⊥i
L ¼
Z d2p⊥
ð2πÞ2p2⊥Cðp⊥Þ: (1)
A leading-order (LO) perturbative evaluation of ˆq was carried out in Ref. [5] and later extended to the next-to-leading order (NLO) in Ref. [6]. However, as is well known, perturbative computations in thermal QCD cannot be pushed to arbitrarily high order, due to the presence of infrared divergences which reveal the intrinsically non-perturbative nature of interactions for long-wavelength modes. In addition, at the temperatures probed at RHIC
(and at LHC) the accuracy of LO or NLO perturbative computations may be questioned.
The gauge-string duality [7] provides a framework in which the phenomenon can be studied nonperturbatively, in the strong-coupling limit and in the large-N approximation. Following this approach, a holographic computation of ˆq (for a light hard parton) was presented in Ref. [8]; for related studies, see also Refs. [9]. However, since the holographic dual of QCD is not known exactly, these types of computations are usually carried out for models (e.g., the N ¼ 4 supersymmetric Yang-Mills theory), which are only expected to reproduce the features of QCD qualitatively or semiquantitatively.
Lattice simulations are the standard tool for nonperturba-tive, first-principle computations in QCD, but since they are based on the regularization of the theory in a Euclidean spacetime, they are not well suited for real-time phenomena, and typically require some analytical continuation. How-ever, a closer inspection of the problem reveals that con-tributions from the parametrically different (“hard” OðπTÞ, “soft” OðgTÞ, and “ultrasoft” Oðg2T=πÞ) energy scales of
the QGP can be disentangled from each other. In particular, following an idea first proposed in Ref. [6], and later discussed also in Refs.[10–14], it is possible to show that the contribution to jet quenching from soft modes can be directly evaluated in lattice simulations of a purely bosonic, dimensionally reduced effective theory (electrostatic QCD, or EQCD[15]). The latter, defined by the Lagrangian
L ¼1
4FaijFaijþ TrððDiA0Þ2Þ þ m2ETrðA20Þ þ λ3ðTrðA20ÞÞ2;
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is three-dimensional SUð3Þ Yang-Mills theory coupled to an adjoint scalar field A0. The values of its parameters (the dimensionful, three-dimensional gauge coupling gE and
the coefficients for the quadratic and quartic terms in the scalar potential) are fixed by a matching procedure, to reproduce the physics of high-temperature QCD.
Lattice formulation.—The lattice regularization of the theory described by the Lagrangian density(2)is straight-forward (see Ref. [16] and references therein). We chose parameter values corresponding to QCD with nf¼ 2 light
quark flavors, at two temperatures T≃ 398 MeV and 2 GeV, respectively, equal to about twice and ten times the deconfinement temperature.
The contribution to jet quenching from soft QGP modes can be extracted from the two-point correlation function of long light-cone Wilson lines (stretching, for example, along the x3− t ¼ const direction at fixed x1 and x2, and separated by a distance r along the x1 direction), which can be made gauge-invariant by“closing the loop” with two transverse parallel transporters, and taking the trace over color indices [10]. In the lattice regularization of EQCD, each lightlike side of this loop becomes a“staircase” path, built from products of unitary parallel transporters U3ðxÞ over one lattice spacing unit a to the x3 direction, and matrices HðxÞ ¼ exp½−ag2EA0ðxÞ, which are EQCD ana-logues of the parallel transporters along paths of length a in the real-time direction. Note that HðxÞ is Hermitian (rather than unitary). Suppressing the coordinate dependence, the lattice operator corresponding to a single light-cone Wilson line is then: L3¼QðU3HÞ. Denoting the path-ordered product of gauge links U1ðxÞ along the transverse direction as L1, the original light-cone Wilson loop can be repre-sented in lattice EQCD as a“decorated” Wilson loop (see Fig.1) defined as
Wðl; rÞ ¼ TrðL3L1L−13 L†1Þ; (3) where l denotes the length of the loop along the x3 direction. Wðl; rÞ enjoys well-defined renormalization
properties [17]. We compute the hWðl; rÞi expectation values with a multilevel algorithm [18] and extract the coordinate-space expression for the differential transverse collision kernel,
VðrÞ ¼ − lim
l→∞
1
llnhWðl; rÞi; (4)
which equals the transverse Fourier transform of−Cðp⊥Þ (up to an additive term, which is inessential to our discussion). Note that the presence of “real time” in Wðl; rÞ implies that this operator is actually very different from a usual spatial Wilson loop. In particular, perturbation theory predicts a partial cancellation between the gauge and scalar propagators [5,6,12,19]. In turn, this implies that VðrÞ starts from zero and is a decreasing function at very small r, while at distances larger than the inverse of the Debye mass mD it includes a linear term of positive slope
7g4
ECfCa=ð64πÞ, with Cf¼ 4=3, Ca¼ 3 [6,12].
Results.—Using Eq. (4), we extracted the differential transverse collision kernel V as a function of the separation r between the lightlike Wilson lines, at the two temper-atures that we investigated. Our simulation results exhibit good scaling properties: data obtained from lattices of different spacing a (recall that, in three-dimensional SUð3Þ lattice gauge theory, β ¼ 6=ðag2EÞ) fall within a narrow band. This indicates that discretization effects are under control, and that our data allow us to obtain a reliable extrapolation to the continuum limit.
Since the jet quenching parameter is the second moment of Cðp⊥Þ, the soft contribution to ˆq is encoded in the
curvature of VðrÞ. Following a procedure similar to the one used in Ref. [11], we fit V=g2E to a functional form including linear, quadratic, and quadratic-times-logarithmic terms in rg2E, i.e., c1rg2Eþ c2ðrg2EÞ2þ c3ðrg2EÞ2lnðrg2EÞ, and estimate the contribution to ˆq from soft modes as f4c2þ 2c3½2 þ γ þ 4 lnðr0g2E= ffiffiffi 2 p Þgg6 E, where γ is the
Euler-Mascheroni constant and r0denotes Sommer’s scale [20], with r0g2E≃ 2.2[21]. At the two temperatures consid-ered, we find that the total contribution toˆq from soft modes is significantly larger than the perturbative NLO expectation,
ˆqNLO ¼ g4T2m DCfCa
3π2þ 10 − 4 ln 2
32π2 ; (5)
(where, at this order, mD ¼ mEþ Oðg2TÞ, with mE¼
gT ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðCaþ nftfÞ=3
p
and tf¼ 1=2). The fact that
contribu-tions beyond NLO (which are intrinsically nonperturbative) are large is not surprising: at those temperatures also the Debye mass mDreceives a numerically large nonperturbative
correction[22], which dominates over the leading perturba-tive term (mE). In fact, it is interesting to note that, when
plotted in terms of the nonperturbatively evaluated Debye mass, our lattice results for VðrÞ become compatible with the theoretical NLO curve[6,12], as seen in Fig.2(which also shows that this rescaling in terms of the Debye mass makes
L1 L1
L3
H
r
L3−1
FIG. 1 (color online). The “decorated” Wilson loop Wðl; rÞ describing a two-point correlation function of light-cone Wilson lines involves Hermitian parallel transporters HðxÞ along the real-time direction.
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the data sets at the two different temperatures compatible with each other: this suggests that, essentially, the temperature dependence of V is inherited from mD). Physically reasonable
values of the coupling g2∼ 2.6[23]for RHIC temperatures (at which the LO contribution to the jet quenching parameter is known to be subdominant[6]) then lead us to estimate ˆq ≃ 6 GeV2=fm. This value is in the same ballpark as those
obtained from holography [8,9], as well as from certain phenomenological models [24] (although the latter are somewhat dependent on the details of the approach that is used [25], and more recent studies tend to favor smaller values[26]).
Conclusions.—In this Letter, we presented a nonpertur-bative investigation of the momentum broadening of a hard parton (specifically, a light quark) in the quark-gluon plasma. Although Monte Carlo simulations on a Euclidean lattice are generally ill-suited for studying phenomena involving real-time dynamics (because they require some analytical continuation), in the present work, following an idea origi-nally proposed in Ref.[6], we extracted the contribution to the jet quenching parameter from soft QGP modes, with momenta OðgTÞ, in a high-precision numerical study of a dimensionally reduced, Euclidean and purely bosonic effec-tive theory (EQCD). Recent works discussing related ideas include Refs.[10–14], while a different approach to study jet quenching on the lattice was suggested in Ref.[27].
Our nonperturbative estimate of the soft contribution toˆq in the dimensionally reduced effective theory is obtained from the numerical evaluation of a lattice operator describ-ing (a gauge-invariant version of) the two-point correlator of light-cone Wilson lines. Our results give direct access to VðrÞ, which is related to the Fourier transform of the collision kernel Cðp⊥Þ. While, by construction, our effec-tive theory approach misses the contribution to ˆq from hard
thermal modes with momenta OðπTÞ (which, however, can be reliably estimated perturbatively, and is numerically subdominant), we remark that it does so in a well-defined way, consistent with the modern theoretical approach to thermal QCD [15,23]. Although the separation between hard, soft and ultrasoft scales may be partially blurred at temperatures accessible to present-technology experiments, in this work we presented a first concrete attempt to provide a quantitative, first-principle estimate forˆq that is free from uncontrolled approximations. We found that soft contribu-tions to ˆq are significantly larger than the perturbative prediction up to NLO. However, this mismatch can be accommodated, by expressing the results in terms of the nonperturbatively evaluated Debye mass. Our final result forˆq at RHIC temperatures is about 6 GeV2=fm (with total uncertainty around 15%–20%). While this estimate should be taken cum grano salis, given that the very definition ofˆq is affected by intrinsic ambiguities (see Ref. [11] for a discussion), we stress that all sources of uncertainty related to the lattice regularization are under control, and system-atically improvable. In particular, our simulations were carried out with large and fine lattices, so finite-volume and finite-cutoff effects are small. Finally, a recent classical lattice gauge theory study[13] finds conclusions that are compatible with ours.
To extend this study, the determination of VðrÞ could be refined using an improved formulation of the lattice action, for which a multilevel algorithm has been recently pro-posed[28]. We also plan to repeat the present calculation at several different temperatures T, in order to study the dependence ofˆq on T. As the temperature is increased, the QGP should interpolate between a regime dominated by nonperturbative physics, and one in which it becomes weakly coupled. It would also be interesting to study the 0 2 4 6 8 10 12 14 16 18 20 22 24 r mD 0 0.1 0.2 0.3 0.4 0.5 0.6 V / mD β = 12 β = 14 β = 16 β = 18 β = 24 β = 32 β = 40 β = 54 β = 67 β = 80 continuum extrapolation NLO PT 0 2 4 6 8 10 12 14 16 18 20 22 24 r mD 0 0.1 0.2 0.3 0.4 0.5 0.6 V / m D β = 12 β = 14 β = 16 β = 18 β = 24 β = 32 β = 40 β = 54 β = 67 β = 80 continuum extrapolation NLO PT
FIG. 2 (color online). The coordinate-space collision kernel V, extracted from expectation values of Wðl; rÞcomputed nonperturbatively in our EQCD simulations. Symbols of different colors correspond to different lattice spacings, while the dashed black line (and the gray band) indicate the continuum limit (and the associated uncertainty). The lhs panel shows results for T≃ 398 MeV, the one on the rhs for T≃ 2 GeV.BothV andrareshownintheappropriateunitsoftheDebyescreeningmassmD, as evaluated nonperturbatively in Ref.[22].
The NLO perturbative prediction[6,12]is also shown (solid black line).
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dependence of ˆq on the number of color charges N, in particular in the large-N limit[29], which provides insight into many properties of real-world QCD (see Refs. [30]) and plays a crucial role in all holographic computations. Lattice studies in both four [31]and three[32] spacetime dimensions show that static equilibrium quantities charac-terizing the QGP have very mild (nontrivial) dependence on N. It would be interesting to see if this also holds for quantities involved in real-time dynamics.
This work is supported by the Academy of Finland, project 1134018, by the Spanish MINECO’s “Centro de Excelencia Severo Ochoa” program under Grant No. SEV-2012-0249, by the German DFG (SFB/TR 55) and partly by the European Community under the FP7 program HadronPhysics3. Part of this work was carried out during the “Heavy quarks and quarkonia in thermal QCD” at ECT⋆ in Trento, Italy. Part of the numerical simulations was carried out at the Finnish IT Center for Science (CSC) in Espoo, Finland. We thank F. Bigazzi, D. Bödeker, S. Caron-Huot, J. Ghiglieri, A. Kurkela, M. Laine, B. Müller, A. Mykkänen, C. Salgado, A. Vairo and X.-N. Wang for helpful comments and discussions.
*
marco.panero@inv.uam.es †kari.rummukainen@helsinki.fi
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